As by now you should fully understand from working with the Poisson
equation, one very general way to solve inhomogeneous partial
differential equations (PDEs) is to build a Green's
function11.1 and write the solution as an
integral equation.

Let's very quickly review the general concept (for a further discussion
don't forget WIYF
,MWIYF). Suppose is a general (second order)
linear partial differential operator on e.g.
and one wishes to
solve the inhomogeneous equation:

(11.26)

for .

If one can find a solution
to the associated
differential equation for a point source function11.2:

(11.27)

then (subject to various conditions, such as the ability to interchange
the differential operator and the integration) to solution to this
problem is a Fredholm Integral Equation (a convolution of the
Green's function with the source terms):

(11.28)

where
is an arbitrary solution to the associated
homogeneous PDE:

(11.29)

This solution can easily be verified:

(11.30)

(11.31)

(11.32)

(11.33)

(11.34)

(11.35)

It seems, therefore, that we should thoroughly understand the ways
of building Green's functions in general for various important PDEs.
I'm uncertain of how much of this to do within these notes, however.
This isn't really ``Electrodynamics'', it is mathematical physics, one
of the fundamental toolsets you need to do Electrodynamics, quantum
mechanics, classical mechanics, and more. So check out Arfken, Wyld,
WIYF
, MWIYFand we'll content ourselves with a very quick review of
the principle ones we need: